Directions/location 离，旁边，外

[AlexanderH]

I'm having a bit of trouble understanding what the text book is trying to teach me here.

I'm just interested in basics sentences, e.g. There are many students at the school. My pen is on the desk. My homework is at home. The sports field is on the side of the library.

The problem is the book first goes into detail about 书上，学校离，电视离。 These are good basic phrases I need to know.

Then, the book mentions that the optional 的 in sentences like 图书馆(的)旁边。 Near the library. But can you also say 图书馆旁?

Comparing 教师外面。 vs 教师外。 Are those both grammatically correct and fully legitimate sentences?

What about 桌子的上头。 v s 桌子上。 The same meaning?

How are most directional commands / statements given in colloquial speech?

Sorry -- a bit confused on what can and cannot be left out here.

[Daan]

The problem is the book first goes into detail about 书上，学校离，电视离。 These are good basic phrases I need to know.

The suffix meaning 'inside of' is lǐ 里 (third tone), not lí 离 (second tone). I assume you intended to write xuéxiàolǐ 学校里 and diànshìlǐ 电视里

Then, the book mentions that the optional 的 in sentences like 图书馆(的)旁边。 Near the library. But can you also say 图书馆旁?

No, but you can say túshūguǎn pángbiān 图书馆旁边. The de 的 is optional, not the biān 边.

Comparing 教师外面。 vs 教师外。 Are those both grammatically correct and fully legitimate sentences?

Neither of them is a sentence: that is, they cannot be used on their own. Rather, they're phrases you can combine with other phrases to form sentences. Also, you've written jiàoshī 教师 which means 'teacher'. I assume you intended to write jiàoshì 教室 'classroom'? In that case, you could expect to hear both 教室外面 and 教室外.

What about 桌子的上头。 v s 桌子上。 The same meaning?

I suppose you could say so, yes, but there are always some differences in usage and meaning when you're discussing two different words. The same applies to 教室外面 and 教室外, of course.

I hope this helps a bit If you're still confused, why don't you try making up some simple sentences and posting them here? We'll be happy to correct them and explain this sometimes confusing part of Chinese grammar further